Induced subgraphs of zero-divisor graphs

G. Arunkumar; Peter J. Cameron; T. Kavaskar; T. Tamizh Chelvam

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arXiv · 2207.11741

Induced subgraphs of zero-divisor graphs

G. Arunkumar·Peter J. Cameron·T. Kavaskar·T. Tamizh Chelvam

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Abstract

The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with $a$ and $b$ adjacent if $ab=0$. We show that the class of zero-divisor graphs is universal, in the sense that every finite graph is isomorphic to an induced subgraph of a zero-divisor graph. This remains true for various restricted classes of rings, including boolean rings, products of fields, and local rings. But in more restricted classes, the zero-divisor graphs do not form a universal family. For example, the zero-divisor graph of a local ring whose maximal ideal is principal is a threshold graph; and every threshold graph is embeddable in the zero-divisor graph of such a ring. More generally, we give necessary and sufficient conditions on a non-local ring for which its zero-divisor graph to be a threshold graph. In addition, we show that there is a countable local ring whose zero-divisor graph embeds the Rado graph, and hence every finite or countable graph, as induced subgraph. Finally, we consider embeddings in related graphs such as the $2$-dimensional dot product graph.

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BibTeX RIS

G. Arunkumar, Peter J. Cameron, T. Kavaskar, T. Tamizh Chelvam. 2022-07-24. Induced subgraphs of zero-divisor graphs. https://arxiv.org/abs/2207.11741

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